Motion in a Straight Line

Overview

This lecture covers the fundamentals of motion in a straight line, including definitions, concepts of rest and motion, distance vs displacement, speed and velocity, acceleration, equations of motion, motion under gravity, Galileo’s ratio, and an introduction to graphs and relative motion.

Mechanics & Kinematics

• Mechanics is the branch of physics studying motion and its causes.
• Kinematics studies motion without considering forces (focus of this chapter).
• Motion in a straight line (one-dimensional motion) is the main topic.

Rest, Motion, and Frame of Reference

• Rest: An object is at rest if its position doesn't change with respect to its surroundings.
• Motion: An object is in motion if its position changes with respect to its surroundings.
• The definition of motion/rest depends on the frame of reference (point of view).

Scalars and Vectors

• Scalars: Quantities with only magnitude (e.g., distance, speed).
• Vectors: Quantities with both magnitude and direction (e.g., displacement, velocity).

Distance and Displacement

• Distance: Actual path length traveled (scalar, always positive).
• Displacement: Shortest straight-line distance from start to end point (vector; can be positive, negative, or zero).
• Displacement in circular motion: Use formula Displacement = 2r sin(θ/2).

Speed and Velocity

• Speed: Rate of change of distance (scalar) = total distance / total time.
• Velocity: Rate of change of displacement (vector) = total displacement / total time.
• Uniform speed/velocity: Constant values over equal time intervals; non-uniform if they change.
• Average speed: Total distance / total time.
• Average velocity: Total displacement / total time.

Acceleration

• Acceleration: Rate of change of velocity = (final velocity - initial velocity)/time.
• Positive acceleration increases speed; negative acceleration (retardation) decreases speed.
• Instantaneous velocity/acceleration: Value at a specific instant (use derivatives: dx/dt for velocity, dv/dt for acceleration).

Calculus in Kinematics

• Differentiation: Used to find velocity from displacement, acceleration from velocity.
• Integration: Used to find velocity from acceleration, displacement from velocity.

Equations of Motion (for constant acceleration)

• v = u + at
• s = ut + ½ at²
• v² – u² = 2as

Motion Under Gravity

• Acceleration due to gravity (g) ≈ 9.8 m/s² (downward).
• When dropped: u = 0, a = –g.
• Ball thrown upward: Final velocity at top = 0, acceleration is –g.
• Time to reach max height: t = u/g; total time up and down: 2u/g.
• Maximum height: h = u²/2g.

Galileo’s Ratio

• Distances covered in successive seconds under gravity follow the ratio 1:3:5:7, etc.

Graphs in Kinematics

• Displacement-time graph: Slope gives velocity.
• Velocity-time graph: Slope gives acceleration; area under curve gives displacement.
• Acceleration-time graph: Area under curve gives velocity.
• Impossible graphs include negative distance-time, circular/looped velocity-time, and negative speed-time graphs.

Relative Motion

• Same direction: Relative velocity = difference of velocities.
• Opposite direction: Relative velocity = sum of velocities.
• Total crossing time for two objects: Total distance (including both lengths) / relative velocity.

Key Terms & Definitions

• Mechanics — Study of motion and its causes.
• Kinematics — Study of motion without forces.
• Frame of reference — System used to measure position/motion.
• Scalar — Quantity with only magnitude.
• Vector — Quantity with magnitude and direction.
• Distance — Length of actual path (scalar).
• Displacement — Straight-line change in position (vector).
• Speed — Distance covered per unit time (scalar).
• Velocity — Displacement per unit time (vector).
• Acceleration — Change in velocity per unit time (vector).
• Retardation — Negative acceleration (slowing down).
• Instantaneous value — Value at a specific instant, found by differentiation.
• Relative velocity — Velocity of one object with respect to another.

Action Items / Next Steps

• Practice numerical problems on equations of motion, motion under gravity, and graphs.
• Complete derivations for equations of motion using integration and graphs.
• Review key formulas and concepts, especially displacement in circular motion and Galileo’s ratio.
• Homework: Solve textbook exercises on average speed, relative motion, and motion under gravity.